SUPER EDGE ANTIMAGIC TOTAL LABELING ON DISJOINT UNION OF CYCLE WITH CHORD

A graph with order and size is called , -edge antimagic total ( , -EAT) if there exist integers 0, 0 and a bijection ∶ ∪ → 1, 2, 3, . . . , such that , ∈ , , . . . , 1 , where . An , -EAT labeling of graph is super if 1, 2, . . . , . In this paper we describe how to construct a super , -EAT labeling on some classes of disjoint union from non isomorphic graphs, namely disjoint union of cycle with cycle with chord ∪ μ 1 ∪ μ ∪ .


Introduction
All graphs in this paper are finite, undirected, and simple.A labeling of a graph is any mapping that sends some set of graph elements to a set of numbers (usually to the positive integers).If the domain is the vertex-set or the edge-set, the labelings are called respectively vertex-labelings or edge-labelings.In this paper we deal with the case where the domain is ∪ , and these are called total labeling.We define the edgeweight of an edge ∈ under a total labeling to be the sum of the vertex labels corresponding to vertices , and edge label corresponding to edge .General references for graphtheoretic notions is [12].A general survey of graph labelings is [6].The concept of , -edge antimagic total labeling, introduced by Simanjuntak et al.
In this paper we investigate the existence of super , -edge-antimagic total labelings for disjoint union of non isomorphic graphs.A number of classification studies on super , −EAT (resp., −EAT ) for disjoint union of non isomorphic graphs has been extensively investigated.For instances, some constructions of super , 0 -edge-antimagic total labelings for ∪ and , ∪ , have been shown by Ivančo and Lučkaničová in [7].
In [5]  More results concerning on super edge antimagic total labeling, see for instances in a nice survey paper by Gallian [6].
Now, we will concentrate on the disjoint union of cycle with cycle with chord, denoted by

Some Useful Lemmas
We start this section by a necessary condition for a graph to be super (a, d)-edge-antimagic total, providing a least upper bound for feasible values of d.
Proof.Assume that a , -graph has a super , -edge-antimagic total labeling ∶ 1 which gives the desired upper bound for the difference .
Next, we restate the following lemma that appeared in [8].

Conclusion
We have lemma and theorem from bijective function of super (a, d)-edge antimagic total labeling on disjoint union of cycle with chord:  ------------------- and (in short, and ) stand for the vertex-set and edge-set of graph , respectively.Let , (in short, ) denote an edge connecting vertices and in .Then, let order | | in denoted by and size | | in denoted by .

= 2 .Theorem 2 , 1 -
It is not difficult to see that the set of : Thus 2 is a super , 2 -edge antimagic total labeling.This concludes the proof.The disjoint union of ∪ edge antimagic total labeling for µ ≥ 1,(µ − 1) ≤ m ≤ µ, n ≥ 7 and n odd.